38. In the gambler's ruin problem of Section 5, suppose the gambler's fortune is presently /, and suppose that we know that the gambler's fortune will eventually reach Í (before it goes to 0). Given this information, show that the probability he wins the next gamble is PV - (Q/P)

l+1

]

1 - (Q/PY '

i + 1 2/ '

Hint: The probability we want is P{Xn+l = i + 1 \Xn = i, lim Xm = N]

_ P[Xn+l = 1 + 1 , hmmXm = N\Xn = i]

P{\immXm = N\Xn = i]